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2 WAVELETS FOR THE ANALYSIS, ESTIMATION, AND SYNTHESIS OF SCALING DATA P. A BRY AND P. F LANDRIN CNRS UMR 5672, E  cole Normale SupeÂrieure de Lyon, Laboratoire de Physique, 69 364 Lyon Cedex 07, France M. S. T AQQU Department of Mathematics, Boston University, Boston, MA 02215-2411 D. V EITCH Software Engineering Research Centre, Carlton, Victoria 3053, Australia 2.1 THE SCALING PHENOMENA 2.1.1 Scaling Issues in Traf®c The presence of scaling behavior in telecommunications traf®c is striking not only in its ubiquity, appearing in almost every kind of packet data, but also in the wide range of scales over which the scaling holds (e.g., see Beran et al. [18], Leland et al. [43], and Willinger et al. [78]). It is rare indeed that a physical phenomenon obeys a consistent law over so many orders of magnitude. This may well extend further, as increases in network bandwidth over time progressively ``reveal'' higher scales. While the presence of scaling is now well established, its impact on teletraf®c issues and network performance is still the subject of some confusion and uncertainty. Why is scaling in traf®c important for networking? It is clear, as far as modeling of the traf®c itself is concerned, that a feature as prominent as scaling 39 Self-Similar Network Traf®c and Performance Evaluation, Edited by Kihong Park and Walter Willinger ISBN 0-471-31974-0 Copyright # 2000 by John Wiley & Sons, Inc. Self-Similar Network Traf®c and Performance Evaluation, Edited by Kihong Park and Walter Willinger Copyright # 2000 by John Wiley & Sons, Inc. Print ISBN 0-471-31974-0 Electronic ISBN 0-471-20644-X should be built into models at a fundamental level, if these are to be both accurate and parsimonious. Scaling, therefore, has immediate implications for the choice of classes of traf®c models, and consequently on the choice, and subsequent estimation, of model parameters. Such estimation is required for initial model veri®cation, for ®tting purposes, as well as for traf®c monitoring. Traf®c modeling, however, does not occur in isolation but in the context of performance issues. Depending on the performance metric of interest, and the model of the network element in question, the impact and therefore the relevance of scaling behavior will vary. As a simple example, it is known that, in certain in®nite buffer ¯uid queues fed by long-range-dependent (LRD) on=off sources, the stationary queueing distribution has in®nite mean, a radically nonclassical result. Such in®nite moments disappear, however, if the buffer is ®nite, intuitively because a ®nite reservoir cannot ``hold'' long memory. The long-range dependence of the input stream will strongly affect the over¯ow loss process but cannot seriously exacerbate the conditional delay experienced by packets that are not lost, as this is bounded by the size of the buffer. The importance of scaling in the performance sense, apart from being as yet unknown in a great many cases, is therefore context dependent. We focus here on the fundamental issues of detection, identi®cation, and measurement of scaling behavior. These cannot be ignored even if one is interested in performance questions that are not directly related to scaling. This is because scaling induces nonclassical statistical properties that affect the estimation of all parameters, not merely those that describe scaling. This, in turn, affects the predictive abilities of performance models and therefore their usefulness in practice. The reliable detection of scaling should thus be our ®rst concern. By detecting the absence or presence of scaling, one will know whether the data need be analyzed by using traditional statistics or by using special statistical techniques that take the presence of scaling into account. Here it is vital to be able to distinguish artifacts due to nonstationarities, with the appearance of scaling, from true scaling behavior. Identi®cation is necessary since more than one kind of scaling exists, with differing interpretations and implications for model choice. Finally, should scaling of a given kind be present, an accurate determination of the parameters that describe it must be made. These parameters will control the statistical properties of estimates made of all other quantities, such as the parameters needed in traf®c modeling or quality of service metrics. As a simple yet powerful example of the above, consider a second-order process Xt, which we know to be stationary, and whose mean m X we wish to estimate from a given data set of length n. For this purpose the simple sample mean estimator is a reasonable choice. The classical result is that asymptotically for large n the sample mean follows a normal distribution, with expectation equal to m X , and variance s 2 X =n, where s 2 X is the variance of X . In the case where X is LRD the sample mean is also asymptotically normally distributed with mean m X ; however, the variance is given by 2c r n a =1  aa1=n, where a P0; 1 and c r P0;I are the parameters describing the long-range dependence [17, p. 160]. This expression reveals that the variance of the sample mean decreases with the sample size n at a rate that is slower 40 WAVELETS FOR THE ANALYSIS, ESTIMATION, AND SYNTHESIS OF SCALING DATA than in the classical case. Noting that the ratio of the size of the LRD-based variance to the classical one grows to in®nity with n, it becomes apparent that con®dence intervals based on traditional assumptions, even for a quantity as simple as the sample mean, can lead to serious errors when in fact the data are LRD. We focus here on how a wavelet-based approach allows the threefold objective of the detection, identi®cation, and measurement of scaling to be ef®ciently achieved. Fundamentally, this is due to the nontrivial fact that the analyzing wavelet family itself possesses a scale-invariant feature, a property not shared by other analysis methods. A key advantage is that quite different kinds of scaling can be analyzed by the same technique, indeed by the same set of computations. The semiparametric estimators of the scaling parameters that follow from the approach have excellent propertiesÐnegligible bias and low varianceÐand in many cases compare well even against parametric alternatives. The computational advantages, based on the use of the discrete wavelet transform (DWT), are very substantial and allow the analysis of data of arbitrary length. Finally, there are very valuable robustness advantages inherent in the method, particularly with respect to the elimination of superposed smooth trends (deterministic functions). Another important issue connected with modeling and performance studies concerns the generation of time series for use in simulations. Such simulations can be particularly time consuming for long memory processes where the past exerts a strong in¯uence on the future, disallowing simple approximations based on truncation. Wavelets offer in principle a parsimonious and natural way to generate good approximations to sample paths of scaling processes, which bene®t from the same DWT-based computational advantages enjoyed by the analysis method. This area is less well developed than is the case for analysis, however. 2.1.2 Mapping the Land of Scaling and Wavelets The remainder of the chapter is organized as follows. Section 2.2, Wavelets and Scaling: Theory, discusses in detail the key properties of the wavelet coef®cients of scaling processes. It starts with a brief, yet precise, introduction to the continuous and discrete wavelet transforms, to the multiresolution analysis theory underlying the latter, and the low complexity decomposition algorithm made possible by it. It recalls concisely the de®nitions of two of the main paradigms of scalingÐself-similarity and long-range dependence. The proper- ties of the wavelet coef®cients of self-similar, long-range-dependent, and fractal processes are then given, and it is shown how the analysis of these various kinds of scaling can be gathered into a single framework within the wavelet representation. Extensions to more general classes of scaling processes requiring a collection of scaling exponents, such as multifractals, are also discussed. The aim of Section 2.3, Wavelets and Scaling: Estimation, is to indicate how and why this wavelet framework enables the ef®cient analysis of scaling processes. This is achieved through the introduction of the logscale diagram, where the key analysis tasks of the detection of scalingÐinterpretation of the nature of scaling and estimation of scaling parametersÐcan be performed. Practical issues in the use of 2.1 THE SCALING PHENOMENA 41 the logscale diagram are addressed, with references to examples from real traf®c data and arti®cially generated traces. De®nitions, statistical performance, and pertinent features of the estimators for scaling parameters are then studied in detail. The logscale diagram, ®rst de®ned with respect to second-order statistical quantities, is then extended to statistics of other orders. It is also indicated how the tool allows for and deals with situations=processes departing from pure scaling, such as super- imposed deterministic nonstationarities. Finally, clear connections between the wavelet tool and a number of more classical statistical tools dedicated to the analysis of scaling are drawn, showing how the latter can be pro®tably generalized in their wavelet incarnations. Section 2.4, Wavelet and Scaling: Synthesis, proposes a wavelet-based synthesis of the fractional Brownian motion. It shows how this process can be naturally and ef®ciently expanded in a wavelet basis, allowing, provided that the wavelets are suitably designed, its accurate and computationally ef®cient implementation. Finally, in Section 2.5, Wavelets and Scaling: Perspectives, a brief indication is given of what may lay ahead in the broad land of scaling and wavelets. 2.2 WAVELET AND SCALING: THEORY 2.2.1 Wavelet Analysis: A Brief Introduction 2.2.1.1 The (Continuous) Wavelet Transform The continuous wavelet decom- position (CWT) consists of the collection of coef®cients fT X a; thX ; c a;t i; a P R  ; t P Rg that compares (by means of inner products) the signal X to be analyzed with a set of analyzing functions c a;t u 1  a p c 0 u À t a  ; a P R  ; t P R &' : This set of analyzing functions is constructed from a reference pattern c 0 , called the mother wavelet, by the action of a time-shift operator t t c 0 tc 0 t À t and a dilation (change of scale) operator d a c 0 t1=  a p c 0 t=a: c 0 is chosen such that both its spread in time and frequency are relatively limited. It consists of a small wave de®ned on a support, which is almost limited in time and having most of its energy within a limited frequency band. While the time support and frequency band cannot both be ®nite, there is an interval on which they are effectively limited. The time-shift operator enables the selection of the time instant around which one wishes to analyze the signal, while the dilation operator de®nes 42 WAVELETS FOR THE ANALYSIS, ESTIMATION, AND SYNTHESIS OF SCALING DATA the scale of time (or, equivalently, the range of frequencies) over which it will be observed. The quantity jT X a; tj 2 , referred to as a ``scalogram,'' can therefore be interpreted as the energy content of X around time t within a given range of frequencies controlled by a. In addition to being well localized in both time and frequency, the mother wavelet is required to satisfy the admissibility condition, whose weak form is  c 0 udu  0; 2:1 which shows it is a bandpass or oscillating function, hence the name ``wavelet.'' Wavelets that are often used in practice include the Haar wavelet, the Daubechies wavelets, indexed by a parameter N  1; 2; .; and the Meyer wavelets. The Haar wavelet c 0 u is discontinuous; it equals 1 at 0 u < 1 2 , À1at 1 2 u 1, and 0 otherwise. The Daubechies wavelet with N  1 is in fact that the Haar wavelet, but the other Daubechies wavelets with N > 1 are continuous with bounded support and have N vanishing moments (i.e., they satisfy Eq. (2.5)). The Meyer wavelets do not have bounded support, in neither the time nor frequency domain, but all their moments vanish and they belong to the Schwartz space; that is, they are in®nitely differentiable and decrease very rapidly to 0 as u tends to ÆI. On the condition that the wavelet be admissible, the transform can be inverted: XtC c  T X a; tc a;t t da dt a 2 where C c is a constant depending on c 0 . This reconstruction formula expresses X in terms of a weighted integral of wavelets (acting as elementary atoms) located around given times and frequencies, thereby constituting quanta of information in the time± frequency plane. For a more general presentation of the wavelet analysis see, for example, Daubechies [24]. Because the wavelet transform represents in a plane (i.e., a two-dimensional (2D) space) the information contained in a signal (i.e., one-dimensional (1D) space), it is a redundant transform, which means that neighboring coef®cients in the time±scale plane share a certain amount of information. A mathematical theory, the multi- resolution analysis (MRA), proves that it is possible to critically sample the time± scale plane, that is, to keep, among the fT X a; t, a P R  , t P Rg, only a discrete set of coef®cients while still retaining the total information in X . That procedure de®nes the so-called discrete (or nonredundant) wavelet transform. 2.2.1.2 Multiresolution Analysis and Discrete Wavelet Transform A multireso- lution analysis (MRA) consists of a collection of nested subspaces fV j g jPZ , satisfying the following set of properties [24]: 1.  jPZ V j f0g,  jPZ V j is dense in L 2 R. 2. V j & V jÀ1 . 2.2 WAVELET AND SCALING: THEORY 43 3. XtPV j D X2 j tPV 0 . 4. There exists a function f 0 t in V 0 , called the scaling function, such that the collection ff 0 t À k, k P Z} is an unconditional Riesz basis for V 0 . To understand the signi®cance of these properties, observe that, from Property 1, the V j are approximation subspaces of the space of square integrable functions L 2 R. Property 4 expresses the fact that the set of shifted scaling functions ff 0 t À k, k P Zg form a ``Riesz basis'' for V 0 ; that is, they are linearly independent and span the space V 0 , but they are not necessarily orthogonal nor do they have to be of unit length. Finding such a function f 0 t is hard, but many candidates for f 0 t are known in the literature. Similarly, Properties 3 and 4 together imply that the scaled and shifted functions ff j;k t2 Àj=2 f 0 2 Àj t À k; k P Zg constitute a Riesz basis for the space V j . The multiresolution analysis involves successively projecting the signal X to be studied into each of the approximation subspaces V j : approx j tProj V j Xt  k a X  j; kf j;k t: Since, from Property 2, V j & V jÀ1 , approx j is a coarser approximation of X than is approx jÀ1 . (Note that some authors use the opposite convention and set V j & V j1 : Property 1 moreover indicates that in the limit of j 3I, all information is removed from the signal. The key idea of the MRA, therefore, consists in studying a signal by examining its coarser and coarser approximations, by canceling more and more high frequencies or details from the data. The information that is removed when going from one approximation to the next, coarser one is called the detail: detail j tapprox jÀ1 tÀapprox j t: The MRA shows that the detail signals detail j can be obtained directly from projections of X onto a collection of subspaces, the W j  V j É V jÀ1 , called the wavelet subspaces. Moreover, the MRA theory shows that there exists a function c 0 , called the mother wavelet, to be derived from f 0 , such that its templates fc j;k t2 Àj=2 c 0 2 Àj t À k; k P Zg constitute a Riesz basis for W j : detail j tProj W j Xt  k d X j; kc j;k t: For example, if the scaling function f 0 t is the function that equals 1 if 0 t 1 and 0 otherwise, then the corresponding mother wavelet c 0 u is the Haar wavelet. 44 WAVELETS FOR THE ANALYSIS, ESTIMATION, AND SYNTHESIS OF SCALING DATA Theoretically, this projection procedure can be performed from j 3ÀIup to j 3I. In practice, one limits the range of indices j to j  0; .; J and thus only considers V J & V JÀ1 &ÁÁÁ&V 0 : This means that we restrict the analysis of X to that of its (orthogonal) projection approx 0 t onto the reference space V 0 , labeled as zero by convention, and rewrite this ®ne scale approximation as a collection of details at different resolutions together with a ®nal low-resolution approximation that belongs to V J : approx 0 tapprox J t  J j1 detail j t   k a X J ; kf J ;k t  J j1  k d X  j; kc j;k t: 2:2 If X isinV 0 , one can obviously replace approx 0 by X in the above relation. Except in the case where X actually belongs to V 0 , selecting V 0 implies some unavoidable information loss [11]. This is entirely analogous to the loss induced by the necessary pre®ltering operation involved in Shannon±Whittaker sampling theory to band-limit a process prior to sampling. Note, however, that there is no additional information loss after the initial projection. Varying J simply means deciding if more or less information is written in details as opposed to the ®nal approximation approx J . Since the approx j are essentially coarser and coarser approximations of X , f 0 needs to be a lowpass function. The detail j , being an information ``differential,'' indicates rather that c 0 is a bandpass function, and therefore a small wave, a wavelet. More precisely, the MRA shows that the mother wavelet must satisfy  c 0 t dt  0 [24]. Given a scaling function f 0 and a mother wavelet c 0 , the discrete (or non- redundant) wavelet transform (DWT) consists of the collection of coef®cients Xt3ffa X J ; k; k P Zg;fd X  j; k; j  1; .; J ; k P Zgg: 2:3 These coef®cients are de®ned through inner products of X with two sets of functions: a X  j; khX ; f  j;k i; d X  j; khX ; c  j;k i; 2:4 where c  j;k (resp., c  j;k  are shifted and dilated templates of fc  (resp., c  0 ), called the dual mother wavelet (resp., the dual scaling function), and whose de®nition depends on whether one chooses to use an orthogonal, semiorthogonal, or biorthogonal DWT 2.2 WAVELET AND SCALING: THEORY 45 (e.g., see Daubechies [24]). In Eqs. (2.2) and (2.4), the role of the wavelet and its dual can arbitrarily be exchanged, and similarly for the scaling function and its dual. In what follows this exchange is performed for simplicity of notation. The d X  j; k constitute a subsample of the fT X a; t, a P R  , t P Rg, located on the so-called dyadic grid, d X  j; kT X 2 j ; 2 j k: The logarithm (base 2) of the scale log 2 a  2 j j is called the octave j, and a scale will often be referred to by its corresponding octave. For the sake of clarity, we henceforth restrict our presentation to the DWT (characterized by the d X  j; k, which brings with it considerable computational advantages. However, the funda- mental results based on the wavelet approach hold for the CWT; see Abry et al. [3, 4]. 2.2.1.3 Key Features of the Wavelet Transform In the study of the scaling processes analyzed below, the following two features of the wavelet transform play key roles:  F1: The wavelet basis is constructed from the dilation (change of scale) operator, so that the analyzing family itself exhibits a scale-in-variance feature.  F2: c 0 has a number N ! 1ofvanishing moments:  t k c 0 t dt  0; k  0; 1; 2; .; N À 1: 2:5 The value of N can freely be chosen by selecting the mother wavelet c 0 accordingly. The Fourier transform C 0 n of c 0 satis®es jC 0 nj % jnj N , jnj30 [24]. 2.2.1.4Fast Pyramidal Algorithm In all of what follows, we always assume that we are dealing with continuous time stochastic processes, and therefore that the wavelet (and approximation) coef®cients are de®ned through continuous time inner products (Eq. (2.4)). One major consequence of the nested structure of the MRA consists in the fact that the d X  j; k and the a X  j; k can actually be computed through a discrete time convolution involving the sequence a X  j À 1; k and two discrete time ®lters h 1 and g 1 . The DWT can therefore be implemented using a recursive ®lter-bank-based pyramidal algorithm, as sketched on Fig. 2.1, which has a lower computational cost than that of a fast Fourier transform (FFT) [24]. The coef®cients of the ®lters h 1 and g 1 are to be derived from f 0 and c 0 [24]. The use of the discrete time algorithm to compute the continuous time inner products d X  j; khX ; c j;k i requires an initialization procedure. It amounts to computing an initial discrete time sequence to feed the algorithm (see Fig. 2.1): a X 0; khX ; f  0;k i, which corresponds to the coef®cients of the expansion of the projection of X on V 0 . From a practical point of view, one deals with sampled 46 WAVELETS FOR THE ANALYSIS, ESTIMATION, AND SYNTHESIS OF SCALING DATA versions of X , which implies that the initialization stage has to be approximated. More details can be found in Delbeke and Abry [27] and Veitch and Abry [75]. The fast pyramidal algorithm is not only scalable because of its linear complexity, On for data of length n, but is simple enough to implement on-line and in real time in high-speed packet networks. An on-line wavelet-based estimation method for the scaling parameter with small memory requirements is given by Roughan et al. [62]. 2.2.2 Scaling Processes: Self-Similarity and Long-Range Dependence We can de®ne scaling behavior broadly as a property of scale invariance, that is, when there is no controlling characteristic scale or, equivalently, when all scales have equal importance. There is no one simple de®nition that can capture all systems or processes with this property; rather there are a set of known classes open to h x h x x x xx x x Fig. 2.1 Fast ®lter-bank-based pyramidal algorithm. The DWT can be computed using a fast pyramidal algorithm: that is, given that we have approximation a X j À 1; k at level j À 1, we obtain approximation a X j; k and detail d X j; k at level j by convolving with h 1 and g 1 , respectively, and decimating. The coef®cients of the ®lters h 1 and g 1 are derived from the chosen scaling function and wavelet f 0 and c 0 . The downarrow stands for a decimation by a factor of 2 operation: one drops the odd coef®cients. An initialization step is required to go from the process X to the approximation of order 0: a X 0; k. 2.2 WAVELET AND SCALING: THEORY 47 expansion. In this section we brie¯y introduce the most well known of these, namely, self-similar, self-similar with stationary increments, and long-range-dependent processes. Please note that throughout this chapter we will use the following convention: fx$gx as x 3 a means that lim x3a f x=gx1, and fx%gx as x 3 a means that lim x3a fx=gxC, where C is some ®nite constant. Recall that a process X fXt, t P Rg is self-similar with parameter H > 0 H-ss if X00 and fXct, t P Rg and fc H Xt, t P Rg have the same ®nite- dimensional distributions. Such a process, obviously, cannot be stationary. The process X is H-sssi if it is H-ss and if, in addition, it has stationary increments, that is, if the ®nite-dimensional distributions of its increments fXt  hÀXt, t P Rg do not depend on t.AnH-sssiprocess with H < 1 has zero mean and a variance that behaves as EX 2 ts 2 jtj 2H . The fractional Brownian motion (FBM), for example, is the (unique) Gaussian H-sssi process, which is simply Brownian motion for H  1 2 . Long-range dependence, 1 on the other hand, is associated with stationary processes. A stationary ®nite-variance process X displays long-range dependence if its spectral density G X n satis®es G X n$c f jnj Àa as n 3 0; 2:6 where 0 < a < 1 and where c f is a nonzero constant. 2 Equation (2.6) implies that the autocovariance rkEZ jZ j  k satis®es rk$c r k aÀ1 as k 3I; 2:7 where c r  c f 2G1 À a sinpa=2, G being (here) the Gamma function [17, p. 43]. Equation (2.7) and (2.6) imply that the covariances rk decay so slowly, that  I kÀI rkI, or equivalently, G Z 0I. There is a close relationship between long-range dependence and self-similar processes. Indeed, the increments of any ®nite variance H-sssiprocess have long- range dependence, as long as 1 2 < H < 1, with H and a related through a  2H À 1: 2:8 In particular, fractional Gaussian noise (FGN), which is the increment process of fractional Brownian motion 3 (FBM) [50] with 1 2 < H < 1, has long-range depen- 1 Long-range dependence is sometimes referred to as ``long memory'' or ``second-order asymptotic self- similarity.'' 2 The index f indicates that this constant is in force in the frequency domain. The corresponding constant appearing in the autocovariance is denoted c r . One can also replace these constants by slowly varying functions but for the sake of simplicity, we will not do this here. 3 Discrete standard FGN is the time series XjB H  j  1ÀB H  j, j  0; 1; .; where B H is FBM. Its spectral density satis®es G X ÀnG X n, and because it is a discrete-time sequence, G X n is concentrated on the interval [À 1 2 ; 1 2 . 48 WAVELETS FOR THE ANALYSIS, ESTIMATION, AND SYNTHESIS OF SCALING DATA [...]... j; k† dX … j; k H † % jk À k H jaÀ1À2N ; jk À k H j 3 ‡I: Note that these two properties of the wavelet coef®cients do not rely on an assumption of Gaussianity In P2 above, we used only weak reformulations (setting j ˆ jH ) of P2 SS and P2 LRD Their general versions ( j not necessarily equal to jH ) can be used to formulate a stronger idealization of strict decorrelation: ID1: EdX … j; k† dX … jH ;... Since a P ‰0; 1†, this is in fact always satis®ed H H EdX … j; k† dX … jH ; k H † % j2j k À 2j k H jaÀ1À2N ; j2j k À 2j k H j 3 ‡I: …2:17† Observe that the exponents in P1 LRD and P2 LRD are different from those in P1 SS and P2 SS, respectively 2.2.3.4 WT of Generalized Scaling Processes The results above can be generalized in a straightforward manner to processes that are neither strictly H-sssi nor... to the measurement of generalized forms of the Fano factor, a well-known descriptor of the burstiness of point processes 2.3.1 An Analysis Tool: The Logscale Diagram 2.3.1.1 The Legacy of P1 and P2 Property P2 is the key to the statistical advantages of analysis in the wavelet domain In sharp contrast to the problematic statistical environment in the time domain due to the long-range dependence, nonstationarity,... de®nition 52 WAVELETS FOR THE ANALYSIS, ESTIMATION, AND SYNTHESIS OF SCALING DATA because an H-sssi process (i.e., with 0 < H < 1† is not differentiable, whereas its integrals are) Then, properties P1 SS and P2 SS still hold replacing H by HY The condition for P1 SS becomes N ! p ‡ 1 [10] and can be rewritten as N ! HY [10] We hereafter say that X is an H-sssi… p† process if it is H-ss and has stationary... derived from …n1 ; n2 †; (iv) in the case of a fractal process, a ˆ 2h ‡ 1, expressions for C…a; c0 † can be found in Flandrin and Goncalves [35, 36] and j1 ˆ 1 and j2 is to be Ë Á identi®ed from the data  P2: fdX …j; k†, k P Zg is stationary and no longer exhibits long-range statistical dependences but only short-term residual correlations; that is, it is short-range dependent (SRD) and not LRD, on condition... dependence at small scales Again, this asymptotic recovering of the underlying power law is not a trivial result It would not, for instance, be obtained with periodogram-based estimates [3] and is due to F1  P2 LRD: It can also be shown [3] that the covariance function of any two wavelet coef®cients is controlled by N and therefore can decay much faster than that of the LRD process itself and is no longer... instance, in the study of  turbulence [22, 63] 2.2.3.7 Processes with In®nite Second-Order Statistics: a-Stable Processes The existence of the wavelet coef®cients, the extensions of P0 SS, P1 SS, and P2 SS, to H-sssi processes without second-order statistics, such as a-stable processes, for instance, have recently been obtained [25, 26, 58] (see also, Pesquet-Popescu [57]) but will not be detailed... WAVELETS AND SCALING: ESTIMATION In this section it is shown in detail how the statistical properties of the wavelet detail coef®cients, summarized in the previous section in the form of properties P1 and P2, 56 WAVELETS FOR THE ANALYSIS, ESTIMATION, AND SYNTHESIS OF SCALING DATA can be applied to the related tasks of the detection, identi®cation, and measurement of scaling The estimation of scaling exponents,... consequence of N !1 (F2) In this case, Eq (2.11) reduces to the fundamental result: EdX … j; k†2 ˆ 2j…2H‡1† C…H; c0 †s2 ; Vk; …2:12† „ „ with C…H; c0 † ˆ jtj2H … c0 …u†c0 …u À t† du† dt and s2 ˆ EX …1†2  P2 SS: Using the speci®c covariance structure of an H-sssi process X …t†, namely, EX …t†X …s† ˆ s2 2H fjtj ‡ jsj2H À jt À sj2H g; 2 …2:13† it can be shown [32, 73] that the correlations between wavelet... asymptotically ef®cient (of minimal variance) The variable mj can therefore be thought of as a near-optimal way of concentrating the gross second-order behavior of X at octave j Furthermore, again from P2, the mj are themselves only weakly dependent, so the analysis of each scale is largely decoupled from that at other scales To analyze the second-order dependence of X …t† on scale, therefore, we are . rely on an assumption of Gaussianity. In P2 above, we used only weak reformulations (setting j  j H )ofP2 SS and P2 LRD. Their general versions ( j not. j3I: 2:17 Observe that the exponents in P1 LRD and P2 LRD are different from those in P1 SS and P2 SS, respectively. 2.2.3.4WT of Generalized Scaling